Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-09T05:35:56.456Z Has data issue: false hasContentIssue false
  • English
  • Français

The transition from two-dimensional to three-dimensional planforms in infinite-Prandtl-number thermal convection

Published online by Cambridge University Press:  26 April 2006

Bryan Travis ,
Peter Olson  and
Gerald Schubert
Bryan Travis
Affiliation:
Earth and Space Sciences Division, Los Alamos National Laboratory. Los Alamos, NM 87545, USA
Peter Olson
Affiliation:
Department of Earth and Planetary Sciences, The Johns Hopkins University, Baltimore, MD 21218, USA
Gerald Schubert
Affiliation:
Department of Earth and Space Sciences, University of California. Los Angeles. CA 90024, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of two-dimensional thermal convection in an infinite-Prandtl-number fluid layer with zero-stress boundaries is investigated using numerical calculations in three-dimensional rectangles. At low Rayleigh numbers (Ra < 20000) calculations of the stability of two-dimensional rolls to cross-roll disturbances are in agreement with the predictions of Bolton & Busse for a fluid with a large but finite Prandtl number. Within the range 2 × 104 < Ra [les ] 5 × 105, steady rolls with basic wavenumber α > 2.22 (aspect ratio < 1.41) are stable solutions. Two-dimensional rolls with basic wavenumber α < 1.96 (aspect ratio > 1.6) are time dependent for Ra > 4 × 104. For every case in which the initial condition was a time-dependent large-aspect-ratio roll, two-dimensional convection was found to be unstable to three-dimensional convection. Time-dependent rolls are replaced by either bimodal or knot convection in cases where the horizontal dimensions of the rectangular box are less than twice the depth. The bimodal planforms are steady states for Ra [les ] 105, but one case at Ra = 5 × 105 exhibits time dependence in the form of pulsating knots. Calculations at Ra = 105 in larger domains resulted in fully three-dimensional cellular planforms. A steady-state square planform was obtained in a 2.4 × 2.4 × 1 rectangular box. started from random initial conditions. Calculations in a 3 × 3 × 1 box produced steady hexagonal cells when started from random initial conditions, and a rectangular planform when started from a two-dimensional roll. An hexagonal planform started in a 3.5 × 3.5 × 1 box at Ra = 105 exhibited oscillatory time dependence, including boundary-layer instabilities and pulsating plumes. Thus, the stable planform in three-dimensional convection is sensitive to the size of the rectangular domain and the initial conditions. The sensitivity of heat transfer to planform variations is less than 10%.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 216 , July 1990 , pp. 71 - 91
Copyright
© 1990 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baumgardner, J. R. 1985 Three-dimensional treatment of convective flow in the Earth's mantle. J. Statist. Phys. 39, 501–511.Google Scholar
Bercovici, D., Schubert, G., Glatzmaier, G. & Zebib, A. 1989 Three-dimensional thermal convection in a spherical shell. J. Fluid Mech. 206, 75–104.Google Scholar
Bolton, E. W. & Busse, F. H. 1985 Stability of convection rolls in a layer with stress-free boundaries. J. Fluid Mech. 150, 487–498.Google Scholar
Busse, F. H. 1967a On the stability of two-dimensional convection in a layer heated from below. J. Maths Phys. 46, 140–150.Google Scholar
Busse, F. H. 1967b The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 30, 625–649.Google Scholar
Busse, F. H. 1981 Transition to turbulence in Rayleigh—Bénard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollup), pp. 97–137. Springer.
Busse, F. H. & Whitehead, J. A. 1971 Instability of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305–320.Google Scholar
Christensen, U. 1987 Time-dependent convection in elongated Rayleigh—Bénard cells. Geophys. Res. Lett. 14, 220–223.Google Scholar
Clever, R. M. & Busse, F. H. 1989 Three-dimensional knot convection in a layer heated from below. J. Fluid Mech. 198, 345–363.Google Scholar
Cserepes, L., Rabinowicz, M. & Rosenberg-Borot. C. 1988 Three-dimensional convection in a two-layer mantle. J. Geophys. Res. 93, 12009–12025.Google Scholar
Curry, J. H., Herring, J. R. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Frick, H., Busse, F. H. & Clever, R. M. 1983 Steady three-dimensional convection at high Prandtl number. J. Fluid Mech. 127, 141–153.Google Scholar
Glatzmaier, G. A. 1989 Numerical simulations of mantle convection: time-dependent three-dimensional compressible spherical shell. Geophys. Astrophys. Fluid Dyn. 43, 223–264.Google Scholar
Grötzbach, G. 1982 Direct numerical simulation of laminar and turbulent Bénard convection. J. Fluid Mech. 119, 27–53.Google Scholar
Houseman, G. 1988 The dependence of convection planform on mode of heating. Nature 332, 346–349.Google Scholar
Machetel, P., Rabinowicz, M. & Bernardet, P. 1986 Three-dimensional convection in spherical shells. Geophys. Astrophys. Fluid Dyn. 37, 57–84.Google Scholar
Machetel, P. & Yuen, D. A. 1987 Chaotic axisymmetrical spherical convection and large-scale mantle circulation. Earth Planet. Sci. Lett. 86, 93–104.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260.Google Scholar
Oliver, D. S. & Booker, J. R. 1983 Planform and heat transport of convection with strongly temperature-dependent viscosity. Geophys. Astrophys. Fluid Dyn. 27, 73–85.Google Scholar
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection. J. Fluid Mech. 8, 183–192.Google Scholar
Ramshaw, J. & Dukowicz, J. 1979 APACHE: A generalized-mesh Eulerian computer code for multicomponent chemically reactive fluid flow. Los Alamos Natl. Lab. Rep. LA-7427.Google Scholar
Schnaubelt, M. & Busse, F. H. 1989 On the stability of two-dimensional convection rolls in an infinite Prandtl number fluid with stress-free boundaries. Z. Angew. Math. Phys. 40, 153–162.Google Scholar
Schubert, G. & Anderson, C. A. 1985 Finite element calculations of very high Rayleigh number thermal convection. Geophys. J. R. Astron. Soc. 80, 575–601.Google Scholar
Straus, J. M. 1972 Finite amplitude doubly diffusive convection. J. Fluid Mech. 56, 353–374.Google Scholar
Sweet, R. 1977 A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimensions. SIAM J. Anal. 14, 706–720.Google Scholar
White, D. B. 1988 The planforms and onset of convection with a temperature-dependent viscosity. J. Fluid Mech. 191, 247–286.Google Scholar
Whitehead, J. A. & Parsons, B. 1978 Observations of convection at Rayleigh numbers up to 760,000 in a fluid with large Prandtl number. Geophys. Astrophys. Fluid Dyn. 9, 201–217.Google Scholar
Zebib, A., Goyal, A. K. & Schubert, G. 1985 Convective motions in a spherical shell. J. Fluid Mech. 152, 39–48.Google Scholar